Finite Group Databasebeta

← #7624362

Subgroups — #7624362

Order 512.

1245 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62252C211yesmaximal, characteristic
2C2×#3342C211yesmaximal, characteristic, complemented
3#33920C212yesmaximal, complemented
4#33916C214yesmaximal, complemented
5#33909C212yesmaximal, complemented
6#33895C212yesmaximal, complemented
7#33880C214yesmaximal, complemented
8#33869C212yesmaximal, complemented
9#33805C211yesmaximal, characteristic, complemented
10#33796C211yesmaximal, characteristic, complemented
11#33762C212yesmaximal, complemented
12#33755C212yesmaximal, complemented
13#33673C211yesmaximal, characteristic, complemented
14#33668C211yesmaximal, characteristic, complemented
15#33649C212yesmaximal, complemented
16#33645C212yesmaximal, complemented
17#33488C211yesmaximal, characteristic, complemented
18#3520—21no—
19C2×#556—21no—
20#3342—21no—
21#3342—22no—
22#3342—21no—
23#3342C2211yescharacteristic
24#3342C2212yes—
25#3342C2214yes—
26#3342C2212yes—
27#3336C2212yes—
28#3334C2211yescharacteristic
29#3302—21no—
30#3300—22no—
31#3298—21no—
32D4×Q16C2211yescharacteristic
33#3285C2212yes—
34#3283C2211yescharacteristic
35#3248—21no—
36#3190—22no—
37#3170C2211yescharacteristic
38#3168C2211yescharacteristic
39C2×#500C2211yescharacteristic
40C2×#485—22no—
41C2×#475—21no—
42#3050C2212yes—
43C2×#467C2212yes—
44C2×#467C2212yes—
45C2×C4×Q16C2211yescharacteristic
46C2×C8×D4C2211yescharacteristic
47C2×#427C2212yes—
48#2255C2212yescomplemented
49#2252C2212yescomplemented
50#2243C2212yescomplemented
51#2241C2212yescomplemented
52#2237C2214yescomplemented
53#2237C2214yescomplemented
54#2237C2214yescomplemented
55#2237C2214yescomplemented
56#2236C2212yes—
57#2236C2212yes—
58#2236C2212yes—
59#2236C2214yescomplemented
60#2235C2212yes—
61#2235C2211yescharacteristic
62#2234C2214yescomplemented
63#2233C2212yes—
64#2232C2211yescharacteristic
65#2229C2212yescomplemented
66#2227C2214yescomplemented
67#2225C2212yescomplemented
68#2225C2212yescomplemented
69#2224C2212yescomplemented
70#2220C2214yescomplemented
71#2219C2214yescomplemented
72#2216C2214yescomplemented
73#2216C2214yescomplemented
74#2211C2212yescomplemented
75#2209C2212yescomplemented
76#2189C2214yescomplemented
77#2188C2211yescharacteristic
78#2188C2212yes—
79#2188C2214yescomplemented
80#2185C2211yescharacteristic
81#2176C2214yescomplemented
82C4×Q32C2212yescomplemented
83C4×SD32C2212yescomplemented
84#2169C2212yes—
85#2168C2214yes—
86D4×C16C2212yes—
87#2163C2211yescharacteristic
88#2162C2211yescharacteristic
89#2159C2214yescomplemented
90#2158C2214yescomplemented
91#2141C2214yescomplemented
92#2138C2214yescomplemented
93#2112C2212yes—
94C2×C22⋊C16C2212yes—
95C2×#135C2312yes—
96C22×Q16C2312yes—
97C22×Q16C2311yescharacteristic
98C23×C8C2312yes—
99#556—22no—
100#556—22no—
101#556—24no—
102#556—21no—
103#556—41no—
104#556—42no—
105#556—41no—
106#554—41no—
107#552—22no—
108#551—42no—
109D4×Q8—21no—
110#548—22no—
111#547—21no—
112#547—21no—
113#546—41no—
114#542—41no—
115#539—42no—
116#533—21no—
117C2×C4⋊Q8—21no—
118#529—21no—
119C2×#126—42no—
120C2×#124—41no—
121#525—22no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—22no—
124#520—41no—
125C2×C4×Q8—21no—
126C2×C4×Q8—41no—
127C2×C4×D4D411yescharacteristic
128C2×#117—22no—
129C2×Q32C2312yes—
130C2×Q32C2314yes—
131C2×SD32C2312yes—
132C2×M32C2314yes—
133C22×C16C2314yes—
134#500—21no—
135#500—21no—
136#500C2311yescharacteristic
137#500C2312yes—
138#500C2312yes—
139#499C2311yescharacteristic
140#495C2311yescharacteristic
141#494C2311yescharacteristic
142#488—21no—
143#487—21no—
144#485—22no—
145#485—24no—
146#485—24no—
147#485—24no—
148#485—24no—
149#485—22no—
150#483—22no—
151#480—22no—
152#479—22no—
153#475—22no—
154#475—22no—
155#475—24no—
156#475—21no—
157#474—21no—
158#467—22no—
159#467—22no—
160#467—21no—
161#467—22no—
162#467—22no—
163#467C2312yes—
164#467C2312yes—
165#467C2314yes—
166#467C2314yes—
167#467C2312yes—
168#467C2311yescharacteristic
169#467C2312yes—
170#466C2312yes—
171#466C2311yescharacteristic
172#465—21no—
173#465—22no—
174#465—22no—
175#464—22no—
176#462—21no—
177#461—21no—
178#452—22no—
179#452—22no—
180#451—22no—
181#448—22no—
182C4×Q16—21no—
183C4×Q16—21no—
184C4×Q16—21no—
185C4×Q16C2312yes—
186C4×Q16C2312yes—
187C4×Q16C2312yes—
188C4×Q16C2311yescharacteristic
189C4×SD16—21no—
190C4×D8C2311yescharacteristic
191C8×D4D411yescharacteristic
192C8×D4D412yes—
193C8×D4D411yescharacteristic
194C8×D4C2311yescharacteristic
195C8×D4C2312yes—
196C8×D4C2311yescharacteristic
197#427—22no—
198#427—22no—
199#427C2312yes—
200#427C2314yes—
201#427C2314yes—
202C2×#107—21no—
203C2×#107C2312yes—
204C2×#107C2312yes—
205C2×#107C2311yescharacteristic
206C2×#106—22no—
207C2×C4⋊C8D411yescharacteristic
208#416—22no—
209C2×#103—22no—
210C2×#103—21no—
211C2×#103—22no—
212C2×C22⋊C8D412yes—
213C2×C4×C8C2311yescharacteristic
214#367C2314yes—
215#367C2314yes—
216#367C2312yes—
217#366C2312yes—
218#366C2314yes—
219#366C2314yes—
220#366C2314yes—
221C4⋊C16C2312yes—
222C4⋊C16C2312yes—
223#358C2314yes—
224#358C2314yes—
225#358C2312yes—
226#358C2314yes—
227#358C2314yes—
228#358C2312yes—
229#358C2312yes—
230#358C2314yes—
231#357C2312yes—
232#357C2314yes—
233C22⋊C16C2311yescharacteristic
234C22⋊C16C2312yes—
235C22⋊C16C2314yes—
236C22⋊C16C2311yescharacteristic
237C22⋊C16C2314yes—
238#346C2312yes—
239C4×C16C2312yes—
240C2×#41—22no—
241C2×#41—41no—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—41no—
245C22×D4D811yescharacteristic
246C23×C4C2×D412yes—
247#135—44no—
248C2×Q16—21no—
249C2×Q16—22no—
250C2×Q16—21no—
251C2×Q16—42no—
252C2×Q16—44no—
253C2×Q16C2411yescharacteristic
254C2×Q16C2412yes—
255C2×Q16C2412yes—
256C2×SD16—22no—
257C2×SD16—21no—
258C2×D8C2411yescharacteristic
259C22×C8—44no—
260C22×C8C2×D412yes—
261C22×C8C2×D412yes—
262C22×C8C2×D412yes—
263C22×C8C2×D411yescharacteristic
264C22×C8C2411yescharacteristic
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—41no—
272C4⋊Q8—41no—
273C4⋊Q8—41no—
274#126—44no—
275#126—44no—
276#126—44no—
277#126—44no—
278#126—42no—
279#126—42no—
280#126—41no—
281#125—21no—
282#125—42no—
283#125—41no—
284#124—21no—
285#124—42no—
286#124—42no—
287#124—44no—
288#124—41no—
289#124—42no—
290#123—42no—
291#123—42no—
292#123—41no—
293#123—42no—
294#123—42no—
295C22⋊Q8—21no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—22no—
302C22⋊Q8—42no—
303C22⋊Q8—42no—
304C22⋊Q8—42no—
305C22⋊Q8—42no—
306C22⋊Q8—41no—
307C22⋊Q8—42no—
308C22⋊Q8—41no—
309C22⋊Q8—42no—
310#121—21no—
311#121—22no—
312#121—42no—
313#121—41no—
314#120—42no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—21no—
318C4×Q8—22no—
319C4×Q8—42no—
320C4×Q8—44no—
321C4×Q8—42no—
322C4×Q8—41no—
323C4×Q8—41no—
324C4×Q8—41no—
325C4×Q8—41no—
326C4×D4—21no—
327C4×D4—41no—
328C4×D4—42no—
329C4×D4—42no—
330C4×D4—41no—
331C4×D4—42no—
332C4×D4D811yescharacteristic
333C4×D4D812yes—
334C4×D4D811yescharacteristic
335C4×D4C2×D411yescharacteristic
336C4×D4C2×D411yescharacteristic
337C4×D4C2×D412yes—
338#117—24no—
339#117—24no—
340#117—22no—
341#117—42no—
342#117—42no—
343#117—41no—
344#117—42no—
345#117—41no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—42no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4D811yescharacteristic
355C2×C22⋊C4—42no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4D812yes—
359C2×C42—42no—
360C2×C42—41no—
361C2×C42C2×D411yescharacteristic
362Q32—42no—
363Q32—44no—
364SD32—42no—
365M32—44no—
366C2×C16—44no—
367C2×C16C2412yes—
368C2×C16C2412yes—
369C2×C16C2414yes—
370#107—22no—
371#107—22no—
372#107—21no—
373#107—22no—
374#107—22no—
375#107—21no—
376#107C2414yes—
377#107C2412yes—
378#107C2411yescharacteristic
379#107C2412yes—
380#107C2411yescharacteristic
381#106—24no—
382#106—24no—
383#106—22no—
384#106—21no—
385C4⋊C8C2×D411yescharacteristic
386C4⋊C8C2×D411yescharacteristic
387C4⋊C8C2×D412yes—
388#103—21no—
389#103—22no—
390#103—22no—
391#103—22no—
392#103—22no—
393#103—22no—
394#103—24no—
395#103—24no—
396#103—22no—
397#103—22no—
398#103—21no—
399#102—21no—
400#102—22no—
401#102—21no—
402C22⋊C8C2×D412yes—
403C22⋊C8C2×D412yes—
404C22⋊C8C2×D412yes—
405C22⋊C8C2×D412yes—
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2411yescharacteristic
409C4×C8C2411yescharacteristic
410C24C2×D812yes—
411#41—82no—
412#41—82no—
413#41—84no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—41no—
420C2×Q8—41no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—42no—
424C2×Q8—84no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×D4—21no—
428C2×D4—42no—
429C2×D4—41no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C2×D811yescharacteristic
433C2×D4C2×D811yescharacteristic
434C2×D4C2×D812yes—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4C2×D811yescharacteristic
445C22×C4C2×D811yescharacteristic
446C22×C4C2×D812yes—
447C22×C4C2×D812yes—
448C22×C4C2×D812yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452Q16—22no—
453Q16—22no—
454Q16—22no—
455Q16—24no—
456Q16—42no—
457Q16—81no—
458Q16—82no—
459SD16—81no—
460SD16—82no—
461D8—22no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—41no—
469C2×C8C22×D411yescharacteristic
470C2×C8C22×D411yescharacteristic
471C2×C8C22×D411yescharacteristic
472C2×C8C22×D412yes—
473C2×C8C22×D412yes—
474C2×C8C2511yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—41no—
501C4⋊C4C2×D811yescharacteristic
502C4⋊C4C2×D812yes—
503C4⋊C4C2×D811yescharacteristic
504C22⋊C4—41no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—42no—
518C22⋊C4—41no—
519C22⋊C4C2×D812yes—
520C22⋊C4C2×D812yes—
521C22⋊C4C2×D812yes—
522C22⋊C4C2×D812yes—
523C42—44no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—41no—
530C42C2×D811yescharacteristic
531C42C2×D811yescharacteristic
532C42C22×D411yescharacteristic
533C42C22×D411yescharacteristic
534C16—22no—
535C16—24no—
536C16—22no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23C22×D812yes—
542C23C22×D812yes—
543C23C22×D811yescharacteristic
544Q8—41no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—41no—
550Q8—82no—
551Q8—84no—
552Q8—84no—
553Q8—82no—
554Q8—162no—
555D4—22no—
556D4—22no—
557D4—42no—
558D4—41no—
559D4—41no—
560D4—84no—
561D4—82no—
562D4—162no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—82no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—162no—
588C2×C4—162no—
589C2×C4C22×D812yes—
590C2×C4C22×D811yescharacteristic
591C2×C4C22×D811yescharacteristic
592C2×C4C22×D812yes—
593C2×C4C22×D811yescharacteristic
594C2×C4C22×D811yescharacteristic
595C2×C4C22×D811yescharacteristic
596C2×C4C23×D411yescharacteristic
597C8—21no—
598C8—21no—
599C8—21no—
600C8—22no—
601C8—22no—
602C8#58512yes—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—84no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#341712yes—
616C22C23×D811yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—22no—
622C4—22no—
623C4—81no—
624C4—82no—
625C4—84no—
626C4—82no—
627C4—81no—
628C4—82no—
629C4—82no—
630C4—81no—
631C4#341412yes—
632C4#348511yescharacteristic
633C4#358611yescharacteristic
634C2—22no—
635C2—42no—
636C2—42no—
637C2—161no—
638C2#6169411yescharacteristic
639C2C2×#341411yescharacteristic
640C2C2×#341711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62252C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#3342C2semidirect
3. #33920C2semidirect
4. #33916C2semidirect
5. #33909C2semidirect
6. #33895C2semidirect
7. #33880C2semidirect
8. #33869C2semidirect
9. #33805C2semidirect
10. #33796C2semidirect
11. #33762C2semidirect
12. #33755C2semidirect
13. #33673C2semidirect
14. #33668C2semidirect
15. #33649C2semidirect
16. #33645C2semidirect
17. #33488C2semidirect
48. #2255C22semidirect
49. #2252C22semidirect
50. #2243C22semidirect
51. #2241C22semidirect
52. #2237C22semidirect
53. #2237C22semidirect
54. #2237C22semidirect
55. #2237C22semidirect
59. #2236C22semidirect
62. #2234C22semidirect
65. #2229C22semidirect
66. #2227C22semidirect
67. #2225C22semidirect
68. #2225C22semidirect
69. #2224C22semidirect
70. #2220C22semidirect
71. #2219C22semidirect
72. #2216C22semidirect
73. #2216C22semidirect
74. #2211C22semidirect
75. #2209C22semidirect
76. #2189C22semidirect
79. #2188C22semidirect
81. #2176C22semidirect
82. C4×Q32C22semidirect
83. C4×SD32C22semidirect
89. #2159C22semidirect
90. #2158C22semidirect
91. #2141C22semidirect
92. #2138C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#622522
C2×#33422
#339202
#339162
#339092
#338952
#338802
#338692
#338052
#337962
#337622
#337552
#336732
#336682
#336492
#336452
#334882